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Theorem 1.2 of Deligne-Mumford's 1969 IHÉS paper, "The irreducibility of the space of curves of given genus," is as follows.

If $g \ge 2$ and $C$ is a stable curve of genus $g$ over an algebraically closed field $k$, then $$H^1(C, \omega_{C/k}^{\otimes n}) = (0)$$if $n \ge 2$, and $\omega_{C/k}^{\otimes n}$ is very ample if $n \ge 3$.

I have two questions, as follows.

  1. What is the easiest way to see that the theorem is true?
  2. Do there exist any alternate presentations of the proof in the literature?

Many thanks in advance.

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    $\begingroup$ Here is a related question: mathoverflow.net/questions/113980/…. To me, your Question 1 seems hard to answer in its current form. If I may ask, is there a specific point or points in Deligne--Mumford's proof that is unclear to you? For example, can you see how the proof would go if $C$ were smooth instead of just stable? (I am not trying to be offensive, just to help potential answerers to calibrate their explanations.) $\endgroup$
    – Lazzaro Campeotti
    Aug 19, 2015 at 14:54
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    $\begingroup$ I want to second the comment of potentially dense: what do you mean by "easiest"? You can form the normalization $\widetilde{C}$ of $C$, you can pullback $\omega_{C/k}$ to $\widetilde{C}$, twist by an invertible ideal sheaf, pushforward to $C$, and find an injection of this pushforward into $\omega_{C/k}$ if the ideal sheaf is appropriately chosen. Is that what you are looking for? $\endgroup$
    – Jason Starr
    Aug 21, 2015 at 13:54

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